594220162016engreportzib0------Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priorsThis article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square-integrable random variables can be used to sample such measures. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.urn:nbn:de:0297-zib-5942210.3934/ipi.20170401438-0064Appeared in: Inverse Problems and ImagingnoT. J. SullivanT. J. SullivanZIB-Report16-30enguncontrolledBayesian inverse problemsenguncontrolledheavy-tailed distributionenguncontrolledKarhunen–Loève expansionenguncontrolledstable distributionenguncontrolleduncertainty quantificationenguncontrolledwell-posednessMEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX)PARTIAL DIFFERENTIAL EQUATIONSPROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)STATISTICSNUMERICAL ANALYSISNumerical MathematicsSullivan, Timno-projectUncertainty Quantificationhttps://opus4.kobv.de/opus4-zib/files/5942/stable_bip.pdf5951engreportzib0--2016-05-25--Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse ProblemsThis paper develops a class of meshless methods that are well-suited to statistical inverse problems involving partial differential equations (PDEs). The methods discussed in this paper view the forcing term in the PDE as a random field that induces a probability distribution over the residual error of a symmetric collocation method. This construction enables the solution of challenging inverse problems while accounting, in a rigorous way, for the impact of the discretisation of the forward problem. In particular, this confers robustness to failure of meshless methods, with statistical inferences driven to be more conservative in the presence of significant solver error. In addition, (i) a principled learning-theoretic approach to minimise the impact of solver error is developed, and (ii) the challenging setting of inverse problems with a non-linear forward model is considered. The method is applied to parameter inference problems in which non-negligible solver error must be accounted for in order to draw valid statistical conclusions.1438-0064urn:nbn:de:0297-zib-59513urn:nbn:de:0297-zib-59513T. J. SullivanJon CockayneChris OatesT. J. SullivanMark GirolamiZIB-Report16-31enguncontrolledProbabilistic NumericsenguncontrolledPartial Differential EquationsenguncontrolledInverse ProblemsenguncontrolledMeshless MethodsenguncontrolledGaussian ProcessesenguncontrolledPseudo-Marginal MCMCNumerical MathematicsSullivan, Timno-projectUncertainty Quantificationhttps://opus4.kobv.de/opus4-zib/files/5951/pmm_pde_bip.pdf6023engreportzib0--2016-08-23--Cameron--Martin theorems for sequences of Cauchy-distributed random variablesGiven a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences.1438-0064urn:nbn:de:0297-zib-60230Han Cheng LieHan Cheng LieT. J. SullivanZIB-Report16-40PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)Numerical MathematicsSullivan, Timno-projectLie, Hanhttps://opus4.kobv.de/opus4-zib/files/6023/ZIB_report.pdf6975engreportzib0--2018-08-01--A Shape Trajectories Approach to Longitudinal Statistical AnalysisFor Kendall’s shape space we determine analytically Jacobi fields and parallel transport, and compute geodesic regression. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and reduce the computational expense by several orders of magnitude. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data.
As application example we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative. Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data only.1438-0064urn:nbn:de:0297-zib-69759Esfandiar Nava-YazdaniHans-Christian HegeHans-Christian HegeChristoph von TycowiczT. J. SullivanZIB-Report18-42enguncontrolledshape space, shape trajectories, geodesic regression, longitudinal analysis, osteoarthritisPROBABILITY AND STATISTICSDIFFERENTIAL GEOMETRY (For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx)STATISTICSVisual Data AnalysisVisual Data Analysis in Science and EngineeringTherapy PlanningHege, Hans-ChristianTycowicz, Christoph vonSullivan, TimNavayazdani, EsfandiarECMath-CH15https://opus4.kobv.de/opus4-zib/files/6975/shape_trj.pdf